On a theorem of A. I. Popov on sums of squares

Berndt, Bruce; Dixit, Atul; Kim, Sun; Zaharescu, Alexandru

doi:10.1090/proc/13547

On a theorem of A. I. Popov on sums of squares
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by Bruce C. Berndt, Atul Dixit, Sun Kim and Alexandru Zaharescu PDF

Proc. Amer. Math. Soc. 145 (2017), 3795-3808 Request permission

Abstract:

Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation involving $r_k(n)$ and certain Bessel functions. We provide a proof of this identity for the first time, as well as for another identity, which can be regarded as both an analogue of Popov’s identity and an identity involving $r_2(n)$ from Ramanujan’s lost notebook.

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